Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications
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| Format: | Preprint |
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2024
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| _version_ | 1866916233730850816 |
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| author | Duong, Xuan Thinh Li, Ji Wu, Liangchuan Yan, Lixin |
| author_facet | Duong, Xuan Thinh Li, Ji Wu, Liangchuan Yan, Lixin |
| contents | In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-Δu(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schrödinger operator $\mathcal L=-Δ+V$, where the nonnegative potential $V$ belongs to the reverse Hölder class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces.
Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01791 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications Duong, Xuan Thinh Li, Ji Wu, Liangchuan Yan, Lixin Analysis of PDEs 42B35, 35K15, 42B37 In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-Δu(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schrödinger operator $\mathcal L=-Δ+V$, where the nonnegative potential $V$ belongs to the reverse Hölder class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces. Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates. |
| title | Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications |
| topic | Analysis of PDEs 42B35, 35K15, 42B37 |
| url | https://arxiv.org/abs/2405.01791 |